Linear Feedback Shift Registers for the Uninitiated, Part IV: Easy Discrete Logarithms and the Silver-Pohlig-Hellman Alg

Jason Sachs September 16, 2017

Last time we talked about the multiplicative inverse in finite fields, which is rather boring and mundane, and has an easy solution with Blankinship’s algorithm.

Discrete logarithms, on the other hand, are much more interesting, and this article covers only the tip of the iceberg.

What is a Discrete Logarithm, Anyway?

Regular logarithms are something that you’re probably familiar with: let’s say you have some number \( y = b^x \) and you know \( y \) and \( b \) but...

Linear Feedback Shift Registers for the Uninitiated, Part III: Multiplicative Inverse, and Blankinship's Algorithm

Jason Sachs September 9, 2017

Last time we talked about basic arithmetic operations in the finite field \( GF(2)[x]/p(x) \) — addition, multiplication, raising to a power, shift-left and shift-right — as well as how to determine whether a polynomial \( p(x) \) is primitive. If a polynomial \( p(x) \) is primitive, it can be used to define an LFSR with coefficients that correspond to the 1 terms in \( p(x) \), that has maximal length of \( 2^N-1 \), covering all bit patterns except the all-zero...

Linear Feedback Shift Registers for the Uninitiated, Part 2: libgf2 and Primitive Polynomials

Jason Sachs July 17, 2017

Last time, we looked at the basics of LFSRs and finite fields formed by the quotient ring \( GF(2)[x]/p(x) \).

LFSRs can be described by a list of binary coefficients, sometimes referred as the polynomial, since they correspond directly to the characteristic polynomial of the quotient ring.

Today we’re going to look at how to perform certain practical calculations in these finite fields. I maintain a Python library on bitbucket called...

Linear Feedback Shift Registers for the Uninitiated, Part I: Ex-Pralite Monks and Finite Fields

Jason Sachs July 3, 20171 comment

Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.

— Évariste Galois, May 29, 1832

I was going to call this short series of articles “LFSRs for Dummies”, but thought better of it. What is a linear feedback shift register? If you want the short answer, the Wikipedia article is a decent introduction. But these articles are aimed at those of you who want a little bit deeper mathematical understanding,...

Ten Little Algorithms, Part 6: Green’s Theorem and Swept-Area Detection

Jason Sachs June 18, 20172 comments

Other articles in this series:

This article is mainly an excuse to scribble down some cryptic-looking mathematics — Don’t panic! Close your eyes and scroll down if you feel nauseous — and...

Donald Knuth Is the Root of All Premature Optimization

Jason Sachs April 17, 20172 comments

This article is about something profound that a brilliant young professor at Stanford wrote nearly 45 years ago, and now we’re all stuck with it.


The idea, basically, is that even though optimization of computer software to execute faster is a noble goal, with tangible benefits, this costs time and effort up front, and therefore the decision to do so should not be made on whims and intuition, but instead should be made after some kind of analysis to show that it has net...

Zebras Hate You For No Reason: Why Amdahl's Law is Misleading in a World of Cats (And Maybe in Ours Too)

Jason Sachs February 27, 20171 comment

I’ve been wasting far too much of my free time lately on this stupid addicting game called the Kittens Game. It starts so innocently. You are a kitten in a catnip forest. Gather catnip.

And you click on Gather catnip and off you go. Soon you’re hunting unicorns and building Huts and studying Mathematics and Theology and so on. AND IT’S JUST A TEXT GAME! HTML and Javascript, that’s it, no pictures. It’s an example of an

The Other Kind of Bypass Capacitor

Jason Sachs January 4, 20173 comments

There’s a type of bypass capacitor I’d like to talk about today.

It’s not the usual power supply bypass capacitor, aka decoupling capacitor, which is used to provide local charge storage to an integrated circuit, so that the high-frequency supply currents to the IC can bypass (hence the name) all the series resistance and inductance from the power supply. This reduces the noise on a DC voltage supply. I’ve...

How to Succeed in Motor Control: Olaus Magnus, Donald Rumsfeld, and YouTube

Jason Sachs December 12, 2016

Almost four years ago, I had this insight — we were doing it wrong! Most of the application notes on motor control were about the core algorithms: various six-step or field-oriented control methods, with Park and Clarke transforms, sensorless estimators, and whatnot. It was kind of like a driving school would be, if they taught you how the accelerator and brake pedal worked, and how the four-stroke Otto cycle works in internal combustion engines, and handed you a written...

Round Round Get Around: Why Fixed-Point Right-Shifts Are Just Fine

Jason Sachs November 22, 20163 comments

Today’s topic is rounding in embedded systems, or more specifically, why you don’t need to worry about it in many cases.

One of the issues faced in computer arithmetic is that exact arithmetic requires an ever-increasing bit length to avoid overflow. Adding or subtracting two 16-bit integers produces a 17-bit result; multiplying two 16-bit integers produces a 32-bit result. In fixed-point arithmetic we typically multiply and shift right; for example, if we wanted to multiply some...

My Love-Hate Relationship with Stack Overflow: Arthur S., Arthur T., and the Soup Nazi

Jason Sachs February 15, 201551 comments

Warning: In the interest of maintaining a coherent stream of consciousness, I’m lowering the setting on my profanity filter for this post. Just wanted to let you know ahead of time.

I’ve been a user of Stack Overflow since December of 2008. And I say “user” both in the software sense, and in the drug-addict sense. I’m Jason S, user #44330, and I’m a programming addict. (Hi, Jason S.) The Gravatar, in case you were wondering, is a screen...

Adventures in Signal Processing with Python

Jason Sachs June 23, 201311 comments

Author’s note: This article was originally called Adventures in Signal Processing with Python (MATLAB? We don’t need no stinkin' MATLAB!) — the allusion to The Treasure of the Sierra Madre has been removed, in deference to being a good neighbor to The MathWorks. While I don’t make it a secret of my dislike of many aspects of MATLAB — which I mention later in this article — I do hope they can improve their software and reduce the price. Please note this...

Chebyshev Approximation and How It Can Help You Save Money, Win Friends, and Influence People

Jason Sachs September 30, 201218 comments

Well... maybe that's a stretch. I don't think I can recommend anything to help you win friends. Not my forte.

But I am going to try to convince you why you should know about Chebyshev approximation, which is a technique for figuring out how you can come as close as possible to computing the result of a mathematical function, with a minimal amount of design effort and CPU power. Let's explore two use cases:

  • Amy has a low-power 8-bit microcontroller and needs to compute \( \sqrt{x} \)...

How to Estimate Encoder Velocity Without Making Stupid Mistakes: Part I

Jason Sachs December 27, 201228 comments

Here's a common problem: you have a quadrature encoder to measure the angular position of a motor, and you want to know both the position and the velocity. How do you do it? Some people do it poorly -- this article is how not to be one of them.

Well, first we need to get position. Quadrature encoders are incremental encoders, meaning they can only measure relative changes in position. They produce a pair of pulse trains, commonly called A and B, that look like...

10 Software Tools You Should Know

Jason Sachs May 20, 201215 comments

Unless you're designing small analog electronic circuits, it's pretty hard these days to get things done in embedded systems design without the help of computers. I thought I'd share a list of software tools that help me get my job done. Most of these are free or inexpensive. Most of them are also for working with software. If you never have to design, read, or edit any software, then you're one of a few people that won't benefit from reading this. 

Disclaimer: the "best" software...

Thermistor signal conditioning: Dos and Don'ts, Tips and Tricks

Jason Sachs June 16, 201114 comments

In an earlier blog entry,  I mentioned this circuit for thermistor signal conditioning:

It is worth a little more explanation on thermistor signal conditioning; it's something that's often done poorly, whereas it's among the easiest applications for signal conditioning.

The basic premise here is that there are two resistors in a voltage divider: Rth is the thermistor, and Rref is a reference resistor. Here Rref is either R3 alone, or R3 || R4, depending on the gain...

Ten Little Algorithms, Part 2: The Single-Pole Low-Pass Filter

Jason Sachs April 27, 201512 comments

Other articles in this series:

I’m writing this article in a room with a bunch of other people talking, and while sometimes I wish they would just SHUT UP, it would be...

Important Programming Concepts (Even on Embedded Systems) Part I: Idempotence

Jason Sachs August 26, 20144 comments

There are literally hundreds, if not thousands, of subtle concepts that contribute to high quality software design. Many of them are well-known, and can be found in books or the Internet. I’m going to highlight a few of the ones I think are important and often overlooked.

But first let’s start with a short diversion. I’m going to make a bold statement: unless you’re a novice, there’s at least one thing in computer programming about which you’ve picked up...

10 Circuit Components You Should Know

Jason Sachs November 28, 20111 comment

Chefs have their miscellaneous ingredients, like condensed milk, cream of tartar, and xanthan gum. As engineers, we too have quite our pick of circuits, and a good circuit designer should know what's out there. Not just the bread and butter ingredients like resistors, capacitors, op-amps, and comparators, but the miscellaneous "gadget" components as well.

Here are ten circuit components you may not have heard of, but which are occasionally quite useful.

1. Multifunction gate (

Byte and Switch (Part 1)

Jason Sachs April 27, 201112 comments

Imagine for a minute you have an electromagnet, and a microcontroller, and you want to use the microcontroller to turn the electromagnet on and off. Sounds pretty typical, right?We ask this question on our interviews of entry-level electrical engineers: what do you put between the microcontroller and the electromagnet?We used to think this kind of question was too easy, but there are a surprising number of subtleties here (and maybe a surprising number of job candidates that were missing...